86 BINOMIAL AND NORMAL DISTRIBUTIONS Ch. 4 



of information is essential to any adequate description of a popula- 

 tion, and also is vital when considering sampling problems. 



In order that we may be able to perform certain useful statistical 

 analyses it usually is necessary to assume (after investigation) that 

 the data conform to some general type of frequency distribution, 

 such as the binomial frequency distribution considered in the preced- 

 ing section. In that section a formula was used to determine the 

 frequency distribution for a binomial population when the basic 

 information (n and p) was available. The formulas and the proce- 

 dures for their use are appropriate for discontinuous measurements 

 which fall into only two categories, such as heads and tails. 



Likewise, we need a mathematical formula which is appropriate 

 when continuous measurements (such as weights, heights, and ages) 

 can be expected to conform to what is called a normal frequency dis- 

 tribution. Mathematicians long ago derived the necessary formula, 

 in fact, it has been derived several different ways, all of which — as 

 rigorous derivations — are inappropriate to this book. However, it is 

 possible here, and useful, to show how the normal distribution is re- 

 lated to the binomial frequency distribution. 



As the number of trials (n) is increased the number of ordinates 

 which graphically represent the binomial frequency distribution also 

 becomes greater. As the n increases the discontinuity of the dis- 

 tribution may become less important and less noticeable for many 

 practical purposes. This matter is illustrated in Figures 4.21A, B, 

 and C, for which p = 1/2 and n = 5, 20, and 100, respectively. In 

 Figure 4.21 A our eyes have to search a bit for the actual form of the 

 distribution; for n = 20, the points rather definitely follow a certain 

 symmetrical curve quite well ; and for n = 100, the points of the graph 

 dot out a symmetrical bell-shaped curve quite clearly. To put the 

 matter another way, if the instructor were to ask each member of 

 the class to draw a smooth curve which seemed to the student to fit 

 the points of the figures best, there would be considerable hesitation 

 and disagreement about Figure 4.21A, much less trouble with Figure 

 4.215, and practically unanimous accord concerning the curve needed 

 for Figure 4.21C 



The student will realize that the labor involved in the construction 

 of figures 4.21 A, B, and C becomes increasingly great as n varies from 

 5 to 100. In view of the fact that Figures 4.215 and C are closely 

 approximated by continuous curves, we might hope that a relatively 

 simple formula for a continuous curve might be employed instead of 

 C n , r-p r q l ~ r , or instead of a summation involving this formula. For- 



